density, the H&D curve and gamma

Density is a measure of how much light a negative transmits, or how much light a print reflects. It is
a simple ratio, expressed as a logarithm. If this brings to mind long-forgotten school days, don't worry:
it really is pretty easy to understand.

The reasons why you would want to understand log densities are easy too. First, they provide an
excellent standardized way of measuring of all densities from minimum to maximum. The latter, usually known
as Dmax, provides a good clue to the maximum density a print can provide: the higher
the Dmax, the better the black. Second, they allow you to describe the tonal range
of a negative in a standardized way. A negative with a short tonal range has a low density range, while one
with a long tonal range has a high density range. Third, if you understand
subject brightness ranges (a free module, but also covered in less detail below), negative density
ranges and print density ranges, you can fit your subjects onto the paper, work out the best development
times, and choose the best paper.

An H&D curve plots density against exposure (strictly, the logarithm of the exposure) and is the
fundamental tool of sensitometry.

Gamma is a measure of contrast, which is essential for calculating development times and fitting negatives
onto paper.

St Martin's, Noize

The darkest shadows, in the roof of the arch, have a density of about 2.10:The brightest highlights
to the left of the arch have a tiny but measurable density: about 0.03. In other words they reflect
around one hundred times as much light. They had to be burned in
(paid module) during printing or they 'blew' as in the unmanipulated print above. KowaSIX, 85/2.8,
Maco Cube 400 (roger)

film densities

Think of a 35mm negative. The film base itself is usually grey, even if you scratch off the emulsion,
and a processed emulsion has a certain minimum density ('fog') which is the result of a small proportion
of even the unexposed emulsion being developed. Between the two of them, they will normally absorb about
half the light that falls on the film, and transmit the other half.

This 'film base plus fog' (fb+f) density is therefore the equivalent of a 2x blocking factor. This could
be expressed as a ratio (2x); in stops (1 stop); or as a logarithm of the ratio. The logarithm to base 10
of 2 is near enough 0.30, so that is the density. If you want more accuracy, it is 0.301029995664...
but 0.30 is close enough.

Now think of an exposed part of the film that allows one-third of the light through: a blocking factor
of 3x. Again you could use a simple ratio (3x) or express it in stops (1.5 stops) or use the log density.
The log to base 10 of 3 is 0.47712125472... -- call it 0.48.

The densest part of the film might transmit no more than one-thousandth of the light falling on it:
1000x, 10 stops, log of 1000 = 3.0. And of course there are countless values in between: how about 27x, a
bit under 4-2/3 stops, or log 27 = 1.43136376416... or near enough 1.43. Log densities may not look all
that familiar or convenient, but they are clearly a lot easier to understand than ratios or fractional
stops.

Heiland densitometer

Our Heiland measures both transmission and reflection densities; the switch on the right of the
body sets which. The upper arm is pushed down in contact with the print or negative to take the
reading. The button on top of the arm is used to set the readout to zero: the density figures appear
on red LEDs behind the dark window on the front of the densitometer arm.

Log densities also have the enormous advantage that because they are logarithms, they can be added or
subtracted to give comparative densities, or divided by 0.3 to give you the answer in stops.

Continuing with the same example, you can subtract the fb+f density (0.30) from the maximum density
(3.0) to discover that the image on the film -- the part you are interested in, after all -- has a density
range of 2.70. This is almost exactly 500x or 9 stops, a fairly high density range, but readily achievable
with quite a few films such as Ilford FP4 Plus in the right developer.

To calculate the maximum absorption factor, you take the antilog of 2.7; to find the answer in stops,
you divide the log density by 0.30, which is the logarithm of 2.0x or 1 stop. Densities above fb+f are the
way that negative densities are normally expressed.

Colepeper Memorial, All Saints' Church, Birchington, Kent

It's easier for us to illustrate negative densities with a 4x5 inch negative (this was shot with
a Linhof Technikardan and 210/5.6 Schneider Symmar) simply because the areas of density are bigger and
easier to measure. Rebate density (film base plus fog) is 0.09; 4x5 inch film base is not tinted grey,
of course. There is a tiny bit of texture in the shadows to the left of the coat of arms, a density of
about 0.14 which equates to 0.05 above the 0.9 of fb+f. Densities as low as about 0.03 are printable so
0.05 is a useful density. Now look at the railing below the coat of arms. The grey area at the right end
has a density of 0.60, or 0.51 above fb+f.

The densest area on the right of the memorial (dark on the negative, light on the print) has a
density of 1.99, or 1.90 above fb+f. This is a somewhat contrasty negative (on Ilford FP4 Plus) because it
was developed with an eye to contact printing on printing-out paper, which demands a contrasty negative.
It would need grade 0 or 00 paper if printed conventionally. Generally the maximum printable density
should be about 1.2 above fb+f.

print densities

Print densities work much the same way as film densities, except that you are dealing with reflected
light instead of transmitted. The base for comparison is the brightest white that the paper base can
deliver. An area that reflects half as much as paper-base white has an absorption factor of 2.0 and
therefore a density of 0.3 (the logarithm of 2.0). The maximum density or Dmax of
the best glossy photographic papers is about 2.3, a ratio of near enough 200x or 1/3 stop under 8 stops.
This would be exceptional, though, and you would normally expect something between 2.1 and 2.2, which is
about 125x to 160x or 7 to 7-1/3 stops. Some papers, especially matte papers, deliver maximum densities
of 2.0 or below.

Although these are the maximum absolute density ranges, the 'dynamic range' -- the range across
which you can see texture or detail -- is always less. With a first class print on the finest papers,
you would be lucky to see a dynamic range of 2.0. In other words, if you took one reading in the
lightest areas to show any texture, and another in the darkest areas to show any texture, the log
density range would be 2.0, or 100x, or 6-2/3 stops.

Tankards and key

All densities are taken from the original print. The brightest highlight at the
foot of the left-hand tankard is about 0.02, while that of the lightest part of the wood inside the
handle of the key is about 0.12. The light area beside the touchmark of the centre tankard is below
0.40 while the area just below its 'belt' to the left of the wood is above 1.40. The darkest area
with texture is under the handle of the tankard on the right and runs around 1.90; the absolute
maximum black is around 2.22. Roger used 5x7 inch Ilford Ortho Plus in a Gandolfi Variant with a
210/5.6 Rodenstock Apo-Sironar-N

the H&D curve

If you expose and develop a film or paper under controlled conditions and then plot a graph of density
against the logarithm of the exposure -- a D/log E curve -- you get a curve that looks something like
the illustration below. Because it expresses the exposure characteristics of the material in question,
it is known as a characteristic curve. The D/log E curve is also known as the H&D curve, after
Ferdinand Hurter (1844-1898) and Vero Charles Driffield (1848-1915) who published the original work on
sensitometry in 1890.

The values on the curve below are relative densities, as described above, and the logarithm of the
exposure in lux-seconds. Bar 2 is 1/100 lux-second; 2 is 100 lux-seconds. The exposure range is therefore
about 10,000:1 (the log of 10,000 is 4) and the density range is a bit over 100:1 (the log of 100 is 2).

Initially there is an area where exposure is insufficient to cause any image at all. This is a measure
of the inertia of the emulsion. Once the inertia is overcome, you start to get ever-increasing densities
as you increase the exposure. After this, the curve is divided into three areas.

First, there is the toe, where quite large increases in exposure result in quite modest increases
in density. 'Long toe' materials have (as the name suggests) a long toe and push well; short toe
materials gain contrast very rapidly if you attempt to push them.

Second, there is the so-called straight-line portion, where the relationship between exposure and
density is photographically useful and pretty much constant, though the line is not really very straight:
it is usually a gentle S-curve and may (especially with a mixed emulsion) have lumps and hollows.

Third, there is the shoulder where density has reached its maximum and can go no higher. It takes a
bigger and bigger increase in exposure to get a smaller and smaller increase in density, until eventually
there is no increase in density at all. The maximum density may even decline with very great increases
in exposure. This is true solarization, so called because it was first seen when the sun appeared on a
negative as a clear spot, not a black one as you would expect in a negative.

gamma

The steepness of the H&D curve is a measure of the contrast of the material in use. The average
slope of the curve (here in red) is generally known as the 'gamma'.

You can see that the gamma up to density 2.0 is about 0.8 (density range 2 divided by exposure
range 2.5) which is pretty steep, the sort of thing you might want for alternative processes. For
normal enlarging you would want something between 0.55 and 0.7, so an exposure range of 2.5 (about 300:1
or 8-1/3 stops) should give a density range of between 1.4 (gamma 0.55) and 1.8 (gamma 0.7).

In practice, there are several refinements of gamma such as Kodak's C.I. or Contrast Index or Ilford's
G-bar, usually written as a capital G with a bar over it. These take account of the different shapes of
characteristic curves and the fact that the so-called straight line portion is rarely if ever actually
straight. Although these refinements are sensitometrically important, gamma is good enough for the vast
majority of practical purposes and for understanding the concept.

gamma infinity

Although more development means more contrast, there is a limit to how much contrast any given
film/developer combination can deliver. Sooner or later, fog levels rise faster than maximum density
and overall contrast falls instead of rising. The point at which this happens -- the point of maximum
contrast -- is somewhat grandiosely known as 'gamma infinity'.

Fast films in speed-increasing developers may struggle to reach a gamma infinity of 1, but slow films
in caustic-hydroquinone developers can deliver more contrast than can ever be used for pictorial purposes,
even with 'alternative' processes that require high-contrast negatives.

interpreting manufacturers' curves

On the left above is a pair of characteristic curves for Ilford Pan F; on the right, the same curves
for HP4 (they are very old curves and appear by kind permission of Ilford Ltd.). You can see first of all
that HP4 has much higher fb+f, around 0.45 instead of 0.20 or so. Then you can see that HP4 is a lot faster:
useful density appears with much less exposure. Third, you can see than Pan F is a short toe material where
the contrast builds rapidly when 'pushed' by extra development while HP4 has a long toe which allows easier
pushing. Current materials such as HP5 Plus have even longer toes and push even better.

time-gamma curves

The longer the development time, the higher the contrast or gamma. It is therefore possible to plot
time-gamma curves, also known as time-contrast curves, showing how the contrast of a film increases with
development time. Unfortunately, time-gamma curves are much harder to find than they used to be. This is
partly because they are tedious and time-consuming (and therefore expensive) to determine, and partly
because very few photographers understood what they were for anyway. Also, they must be plotted separately
for every single film/developer combination. But you will see why they can be useful shortly.

On the left are time-contrast curves for Ilford Pan F in three developers with intermittent agitation;
on the right, for Ilford HP4 in ID-11, with both constant (solid line) and intermittent agitation (broken
line). Again they appear by kind permission of Ilford. For an ISO contrast of just under 0.62 you can
see that Pan F needs just under 4 minutes in Microphen, a bit over 6 minutes in ID-11, and over 10 minutes
in Perceptol. You can also see that if you wanted 0.55 with HP4 in ID-11 you would give just under 6
minutes with constant agitation or around 7 minutes with intermittent agitation.

subject brightness ranges

As described
in the free module on subject brightness,
an 'average' subject brightness range is taken to be around 7 stops
or 128:1, a log range of 2.10. This corresponds reasonably closely to
the brightness range of a print, as stated above, so it is possible
to reproduce a 'normal' scene more or less 1:1 in a black and white print.

If the subject
brightness range is longer than this, you will have to sacrifice some
of the highlights or shadows, or compress the brightness range in
some way so that it fits onto the paper. You can do this by
curtailing development (there is a paid module on negative
development technique); or using a softer grade of paper (there
will be another paid module on paper grades); or dodging
and burning (yet another paid module -- sorry).

image brightness range

There is an often-forgotten step between subject brightness range and film contrast, which is the
image brightness range: that is, the brightness range of the image on the ground-glass or indeed on the
film. This is always less than the subject brightness range because of flare.

flare

Flare arises as a result of non-image-forming light that bounces around inside the camera and lens.
This is distributed more or less evenly across the image, but it has very little effect in the highlights
where it is a tiny fraction of the image-forming light falling on the film. In the shadows, on the other
hand, it may be comparable with the strength of the image-forming light. The net effect is a reduction of
the image brightness range via 'filling' of the shadows.

Thus, if the subject brightness range is 128:1, log range 2.1, and the image brightness range is 64:1,
log range 1.8, the camera-lens system has a flare factor of 2 (= 128/64). In practice, a modern
multi-coated prime lens in a large format camera may have a flare factor as near as unity as makes no
odds -- in other words, the image brightness range is very, very close to the subject brightness
range -- while an old box camera with an uncoated lens might have a flare factor of 4.

flare factors with different cameras and lenses

If you use several different cameras, or lenses, or worst of all, several different formats,
your flare factors are quite likely to range from 1 to 2 even if you use first-class equipment.
You can do the sums yourself to see what difference this makes to the necessary gamma.

Tankard and chair

This was shot with an old Vivitar Series 1 90-180 Flat Field, a brilliant lens
in its day and one that still commands a cult following some twenty or thirty years later. It is
however much 'flatter' or less contrasty than the majority of our prime lenses. The choice lies
between increasing the film development times or printing on paper about a grade harder than we
would need with (say) the 90/4 Macro-Elmar. Roger used a Nikon F and Ilford XP2 for this shot,
which is always a little low in contrast, so this print had to be made on grade 4.

flare and enlargers

Exactly similar considerations apply to enlargers, so the image that is formed upon the paper will
once again be of lower contrast than the image on the film (there's no flare in contact printing, obviously).
A flare factor of 2 is quite possible (log = 0.3) so if you are using a paper which will give a full
tonal range with a log exposure range of 1.1 then you need a negative density range of 1.4 to allow for
flare losses here.

film and paper contrast

It might seem logical that both film and paper would be developed to a gamma of 1.0, but this would
give very little flexibility: the films would soon run out of density, and besides, there are the flare
factors mentioned above. In practice, films are normally developed to a gamma of between 0.55 and 0.70.
This means -- logarithms again -- that a subject brightness range of 2.10 was reduced to a negative
density range of anything from 1.16 (gamma 0.55) to 1.47 (gamma 0.70).

Paper contrast, meanwhile, is described in terms of the log exposure range that is required to give a
full density range. Thus an ISO(R) of 100 corresponds to a log exposure range of 1.0. If you can produce
a negative where the subject brightness range is represented by a negative density range of 1.0, it will
'fit' perfectly onto that paper. From this it follows that papers are developed to much higher gammas
than film: anything from about 1.2 for ultra-soft grades to 5 for the hardest grades. In practice papers
are usually developed to completion, i.e. to the highest contrast they can attain, so paper contrast is
essentially a function of emulsion design.

changing gamma to suit the subject

You can see where we are going from here. If a subject with a long brightness range is developed to a
lower gamma, and a subject with a short brightness range to a higher gamma, then both subjects can be
made to give the same negative density range and therefore to print on the same grade of paper.

Let's start with a paper that gives a full range with an exposure range of 1.00 (i.e. 1.00 log
exposure units spans the range from pure white to maximum black, if the paper is developed to completion).
This is a realistic specification for a grade 2 paper.

Now assume an enlarger flare factor of 2, i.e. a log factor of 0.3. The negative density range must
be 1.3 to allow for losses in flare here. In other words, we have to get the subject brightness range -- whatever
it may be -- to 1.3 log density range on the negative.

Now assume a subject brightness range of 2.40 and (once again) a flare factor of 2.0, log = 0.3. The
image brightness range will be 2.1, and in order to get a log density range of 1.3 from an image
brightness range of 2.1 we need a gamma of 1.3/2.1 = 0.62, which by a happy coincidence is very close to
the ISO standard contrast -- see the free module on ISO speeds.

In practice, both camera/lens flare and enlarger flare are likely to be less, and if for example we
are making a contact print from a negative exposed in a large format camera with a negligible flare factor
we need a negative density range of 1.00 from an image brightness range of 2.40 and a gamma as low as
0.42. If the subject brightness range is still greater, the gamma will need to be still lower.

St Clement's, New Romney

The brightness range here is immense. From the dark wood of the beams to the light through the
window above the altar through the arch it is probably over 1000:1 or log range 3.0. Even so, the
solution lies as much in dodging and burning as in anything else. The film exposure was determined by
the need to keep texture in the roof beams; the print exposure was then made as an average, dodging the
beams to keep them from going to dark and burning the light areas so they were not too bright. Roger used
an M-series Leica with a 21/2.8 Kobalux from Adorama, shooting on Ilford XP2 Super because of its long,
straight characteristic curve.

the full sequence

Armed with all this information, it is therefore possible to match just about any subject to just
about any paper grade, provided you know the following:

1

Subject brightness range (easily determined with a spot meter)

2

Flare factor for camera and lens

3

Development time for a standard gamma in a given developer

4

Time-gamma curve for the film and developer in use, obtainable by experiment or from the
manufacturers

5

ISO(R) of the paper, obtainable from the manufacturer

6

Flare factor for the enlarger and its lens

7

Any personal adjustments you need to make to compensate for your own equipment or working techniques.

In practice, headings 2, 6 and 7 mean that rather than wasting hours on formal experiments it is
generally better to rely on trial and error, which eventually becomes consolidated under the heading
'experience', rather than trying to quantify everything.

This is yet another reason why we do not use the Zone System (a free module
gives the rest of the reasons). With one camera, one lens and one enlarger it is tedious enough. Add
further equipment, and to do it properly, you need to carry out separate tests for each enlarger, camera
or lens. In practice, hardly anyone ever does, and they still get good results even when using wildly
disparate equipment. This strongly suggests to us that they are deluding themselves about the degree of
precision that is either attainable or necessary.

Even so, knowing what happens at each stage and what it means is extremely useful, especially if you want to avoid
simplistic or reductionist theories.

the bottom line

Understanding densities, logs and the D/log E curve is essential if you want to understand basic
sensitometry. It is a bit like hard work at first, but once you have grasped the basics, everything
is in your grasp. What is more, you are not at the mercy of gurus or Zone System adherents with their
own private vocabularies: you are using the same internationally agreed vocabulary that film, paper
and developer manufacturers use.

Window, Burgundy

There is a very good range of densities on this print, from the pitch-black of
the windows though a full range of dark and light mid-tones to the white spot of lichen above the
window -- in which there is still texture and detail. There is also a certain amount of psychological
trickery: the brain supplies a great deal of texture that isn't there, because we all know what
stone, weathered wood and rusty iron look like. This is printed on grade 3-1/2 paper from an Ilford
XP2 negative (it was shot in the early 1990s) exposed using a Nikon F and a 70-210/2.8 Sigma Apo
zoom. (Roger)